# Cartesian Product is Small It has been suggested that this page or section be merged into Equivalence of Definitions of Ordered Pair. (Discuss)

## Theorem

Let $a$ and $b$ be small classes.

Then their Cartesian product $a \times b$ is small:

$\map {\mathscr M} {a \times b}$

## Proof

 $\ds a \times b$ $=$ $\ds \set {\tuple {x, y}: x \in a \land y \in b}$ Definition of Cartesian Product $\ds$ $=$ $\ds \set {\tuple {x, y}: \set x \subseteq a \land \set y \subseteq b}$ Singleton of Element is Subset $\ds$ $\subseteq$ $\ds \set {\tuple {x, y}: \set {x, y} \subseteq a \cup b}$ Set Union Preserves Subsets $\ds$ $\subseteq$ $\ds \set {\tuple {x, y}: \tuple {x, y} \subseteq \powerset {a \cup b} }$ Power Set of Subset and Subset Relation is Transitive

So by definition of power set:

$a \times b \subseteq \powerset {\powerset {a \cup b} }$

By Union of Small Classes is Small, $a \cup b$ is small.

By the Axiom of Powers, $\powerset {\powerset {a \cup b} }$ is small.

By Axiom of Subsets Equivalents, $a \times b$ is small.

$\blacksquare$